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You wrote: “Those ad hoc analyses can be useful ways of motivating or justifying an equation…”

For the quadratic term in lunar distances the result as you have given is both correct and fully adequate for all practical purposes. Ad hoc analyses are also absolutely fine however they don’t necessarily provide guidance when one seeks to extend them in a systematic way which was what I was aiming to do for my own interest and understanding.

With regard to my statement: “The problem requires eliminating the azimuth difference, Z” you noted “I don't see the necessity of this, but maybe it's required from a particular point of view?”

I would say it’s my interpretation given the structure of the linear and quadratic terms. The azimuth difference, Z, is absent and the coefficients of the expansion are given terms of, d, and the corner angles of the star and the Moon in the spherical triangle. That is the structure I was aiming to replicate and continue. In principle you could eliminate one of those parameters in favour of Z but then things are no longer nice and symmetric and could probably get pretty ugly.

You asked “I'm curious why you wouldn't derive the series simply by applying the standard two-dimensional Taylor expansion?”

I think that’s equivalent to what I do but ultimately you need to be able to express the coefficients in the Taylor series at all orders just in terms of d and the corner angles. In my case this is accomplished by starting from an exact expression containing only these parameters. Getting a suitable expression that satisfies this requirement is where Cagnoli comes in. Actually looking back at the derivation I do think it is somewhat remarkable how in just the second equation the right hand sides of the cosine rule, the analog formula and Cagnoli’s equation all just naturally appear (in the parentheses).

As to your historical questions: I would like to know the answer to these. I believe I tried to find the source at the time but could not and had to do it myself,